Mani says axioms are not "lies" -- but my point is, are they "true"?
If you are unwilling to say the Axiom of Infinity is "True", does that
mean you would have been unwilling to say that the Prime Number theorem
was "True" in 1945 when only analytical proofs were known, but willing
to say that the Prime Number Theorem was "True" in 1950 after the work
of Erdos and Selberg?
You can't get away with being indifferent to whether an axiom is "True"
unless you are willing to be just as indifferent to whether any therems
proved using that axiom are "True". Where I see a potential "ethical
issue" is when mathematicians talk quite freely about whether
statements like the prime number theorem are "true" but are unwilling
to grant the same status to axioms used to prove those theorems.
(Previous conversation went as follows:)
Shipman:
It seems to me that there is a bit of the "noble lie" here -- because
these finitists (and also the agnostics about infinity) are benefiting
from the use of the Axiom of Infinity by the entire society of
mathematicians, even when they don't use it in their own work, because
the Axiom of Infinity has been so useful in the development of
mathematics as a whole. And of course those skeptics who DO nonetheless
use the axiom are in an even less defensible position.
Mani:
Axioms are not lies. Take it or leave it... of course if you can stand
up with
it.
Shipman:
Does anyone perceive an ethical issue here?
Mani:
No I do not see any. Doing comparisons between two concepts one in a
speculative 'idealist' domain and another in a real domain is more
questionable and unjustifiable. This is a huge subject.